Although the month of Muharram is a sanctified month as a whole, yet, the
10th day of Muharram is the most sacred among all its days. The day is named
'Ashurah'.

According to the Holy Companion Ibn 'Abbas (Radhiallaahu Ánhu). The Holy Prophet (Sallallaahu Alayhi Wasallam), when migrated to Madinah, found that the Jews of Madinah used to fast on the 10th day of Muharram. They said that it was the day on which the Holy Prophet Musa (Moses) (Alayhis salaam) and his followers crossed the Red Sea miraculously and the Pharaoh was drowned in its waters. On hearing this from the Jews, the Holy Prophet (Sallallaahu Alayhi Wasallam) said, "We are more closely rotated to Musa (Alayhis salaam) than you" and directed the Muslims to fast on the day of 'Ashura'. (Abu
Dawood)

It is also reported in a number of authentic traditions that in the
beginning, fasting on the day of 'Ashura' was obligatory for the Muslims. It was later that the fasts of Ramadhan were made obligatory and the fast on the day of 'Ashura' was made optional. Sayyidina 'Aisha (Radhiallaahu Ánha) has said:
"When the Holy Prophet (Sallallaahu Alayhi Wasallam) came to Madinah, he fasted on the day of 'Ashura' and directed the people to fast it. But when the fasts of Ramadhan were made obligatory, the obligation of fasting was confined to Ramadhan and the obligatory nature of the fast of 'Ashura' was abandoned. Whoever so desires should fast on it and any other who so likes can avoid fasting on it." (Sunan Abu Dawud)

However, the Holy Prophet (Sallallaahu Alayhi Wasallam) used to fast on the day of 'Ashura' even after the fasting in Ramadhan was made obligatory. Abdullah ibn Musa (Radhiallaahu Ánhu) reports that the Holy Prophet (Sallallaahu Alayhi Wasallam) preferred the fast of 'Ashura' on the fasts of other days and preferred the fasts of Ramadhaan on the fast of 'Ashura'. (Bukhari and Muslim)

In short, it is established through a number of authentic Hadiths that
fasting on the day of 'Ashura' is Sunnah of the Holy Prophet (Sallallaahu
Alayhi Wasallam) and makes one entitled to a great reward.

According to another Hadith, it is more advisable that the fast of 'Ashura'
should either be prefixed or suffixed by another fast. It means that one
should fast two days: the 9th and 10th of Muharram or the 10th and 11th of it. The reason of this additional fast as mentioned by the Holy Prophet
(Sallallaahu Alayhi Wasallam) is that the Jews used to fast on the day of
'Ashura alone, and the Holy Prophet (Sallallaahu Alayhi Wasallam) wanted to distinguish the Muslim way of fasting from that of Jews. Therefore, he
advised the Muslims to add another fast to that of 'Ashura'.

The Prophet (peace and blessings of Allaah be upon him) said: “For fasting the day of ‘Aashooraa’, I hope that Allaah will accept it as expiation for the year that went before.” (Reported by Muslim, 1976). This is from the bounty of Allaah towards us: for fasting one day He gives us expiation for the sins of a whole year. And Allaah is the Owner of Great Bounty.